For what value of $x$ does $3^{2x^{2}-5x+2} = 3^{2x^{2}+7x-4}$? Express your answer as a common fraction.
If $3^{2x^{2}-5x+2} = 3^{2x^{2}+7x-4}$, then $2x^{2}-5x+2 = 2x^{2}+7x-4$.  We can eliminate the $2x^2$ term from each side and solve $-5x+2=7x-4$ for $x$ to get $x=\boxed{\frac{1}{2}}$.